From Randomness to Order

Mathematical analysis of Schelling segregation

Abstract

A major achievement of the Nobel prize winning economist
Thomas Schelling
was an elegant model of racial segregation, first described in 1969.
In this model, one starts with a random arrangement of two types of objects
(in one or two dimensions) and initiates a stochastic process which rearranges
the objects according to some rules, which induce a probabilistic (nondeterministic) process.
The aim of the model was to understand the large scale segregation observed between black and
white communities in American cities. However the model can be used to interpret other phenomena
such as inter-particle forces.

Since the 1970s there have been a number of studies of this model. However,
these are geared towards experiments (computer simulations) or use modified
(perturbed) versions of the model which make the corresponding Markov process
more regular and suitable for an analysis via methods of statistical mechanics and evolutionary game theory.
The first theoretical analysis of the actual (one dimensional) model appeared recently in a paper by
Brandt, Immorlica, Kamath and Kleinberg, which dealt with a specific case.

For the first time we have rigorously and globally analysed the
one-dimensional Schelling model and have achieved a deeper understanding of
its interesting behaviour, such as situations where an increase in racial tolerance
increases segregation. Along with this theoretical analysis, we have produced a number
of computer simulations which illustrate our theorems. A
picture
produced via such simulations
recently won the
Picturing Science competition of the
Royal Society.

One of our graphical representations is demonstrated with two examples in the above figure.
Here two types of individuals (coloured black and grey) are arranged randomly in a circle in the centre of the large
circle (which is collapsed to a point in the figure). During the stages of the process, individuals swap places with the incentive
to move to a neighbourhood with higher concentration of individuals of their own colour.
A key parameter is the proportion of nodes of a different
colour which any node will tolerate within their neighbourhood.
Once this threshold is passed, they become unhappy and desire to move out.
This is the intolerance is a parameter of the system,
which takes values from 0 to 1 and affects the behaviour of the system significantly.
The main goal of this research is to classify the behaviour of the system (and in particular the final state) according to
the value of the intolerance.
In our graphical representation, each swap is represented with a dot
between the (large) circle and its centre, with colour the latest colour of that position which changed inhabitant.
Moreover the dots are placed at a distance from the centre which is analogous to the time where the change happened.
Hence dots that occur closer to the centre depict swaps that occurred earlier in the process. The outer (large) circle
shows the state of the system at the end of the process.

This 2-dimensional representation allows a record of the entire series of states of the one-dimensional system.
The two pictures correspond to the processes with different intolerance
parameters, 0.485 and 0.49 respectively. In both cases, cascades of swaps of the same type
are clearly visible. On the other hand, such cascades are more restricted in the second figure.
This is a demonstration of a paradox that was discussed empirically by Schelling more than 40 years ago:
in many cases even when people have a preference for a mixed neighbourhood, the system (driven by the incentives of the individual agents) arrives to a high degree of segregation globally. Furthermore, we demonstrate a more extreme version of this observation. We show that in some cases (as the figures suggest) increased tolerance leads to increased segregation.
In our work we have rigorously proved this
phenomenon and specified the conditions that are required for its occurrence.
The outcomes of computer simulations
correspond rather precisely to parts of the theory that we developed in order to prove rigorous results for the
outcome of the process, depending on the parameters.

In this talk I will present our results and illustrate them via a number of computer simulations (pictures and animations).
This is joint work with Richard Elwes (University of Leeds) and
Andy Lewis-Pye (London School of Economics).